A Ferris wheel has a radius of 120.6 meters, and (while spinning) one rotation takes 260 seconds.
Samuel gets into a carriage at the bottom of the Ferris wheel. At \(t=0\), the wheel starts spinning, and it spins continuously for 760 seconds until an abrupt stop. At that moment, Samuel drops his keys, which fall directly down to the ground.
How far, in meters, are the keys from where Samuel entered the carriage? The tolerance is \(\pm 0.1\) meters.
Solution
Since Samuel got on at the bottom of the wheel, he begins at the midline (horizontally). Thus, we can use a stretched, but unshifted, sine wave to model Samuel’s horizontal position over time.
\[x~=~A\cdot\sin(Bt)\]
The amplitude will match the radius of the wheel.
\[A=120.6\]
To get \(B\), the angular frequency, we divide \(2\pi\) by the period (which was given to us).
\[B~=~\frac{2\pi}{P}\]
\[B~=~\frac{2\pi}{260}\]
\[B~\approx~0.0241661\]
Notice,
\[x~=~120.6\cdot\sin(0.0241661\cdot 760)\]
\[x~=~-54.3681095\]
We only care about the absolute value of \(x\).
\[|x| ~=~ 54.3681095\]
Question
A Ferris wheel has a radius of 20.1 meters, and (while spinning) one rotation takes 34 seconds.
Samuel gets into a carriage at the bottom of the Ferris wheel. At \(t=0\), the wheel starts spinning, and it spins continuously for 72 seconds until an abrupt stop. At that moment, Samuel drops his keys, which fall directly down to the ground.
How far, in meters, are the keys from where Samuel entered the carriage? The tolerance is \(\pm 0.1\) meters.
Solution
Since Samuel got on at the bottom of the wheel, he begins at the midline (horizontally). Thus, we can use a stretched, but unshifted, sine wave to model Samuel’s horizontal position over time.
\[x~=~A\cdot\sin(Bt)\]
The amplitude will match the radius of the wheel.
\[A=20.1\]
To get \(B\), the angular frequency, we divide \(2\pi\) by the period (which was given to us).
\[B~=~\frac{2\pi}{P}\]
\[B~=~\frac{2\pi}{34}\]
\[B~\approx~0.1848\]
Notice,
\[x~=~20.1\cdot\sin(0.1848\cdot 72)\]
\[x~=~13.0529437\]
We only care about the absolute value of \(x\).
\[|x| ~=~ 13.0529437\]
Question
A Ferris wheel has a radius of 46.2 meters, and (while spinning) one rotation takes 85 seconds.
Samuel gets into a carriage at the bottom of the Ferris wheel. At \(t=0\), the wheel starts spinning, and it spins continuously for 215 seconds until an abrupt stop. At that moment, Samuel drops his keys, which fall directly down to the ground.
How far, in meters, are the keys from where Samuel entered the carriage? The tolerance is \(\pm 0.1\) meters.
Solution
Since Samuel got on at the bottom of the wheel, he begins at the midline (horizontally). Thus, we can use a stretched, but unshifted, sine wave to model Samuel’s horizontal position over time.
\[x~=~A\cdot\sin(Bt)\]
The amplitude will match the radius of the wheel.
\[A=46.2\]
To get \(B\), the angular frequency, we divide \(2\pi\) by the period (which was given to us).
\[B~=~\frac{2\pi}{P}\]
\[B~=~\frac{2\pi}{85}\]
\[B~\approx~0.0739198\]
Notice,
\[x~=~46.2\cdot\sin(0.0739198\cdot 215)\]
\[x~=~-12.6134916\]
We only care about the absolute value of \(x\).
\[|x| ~=~ 12.6134916\]
Question
A Ferris wheel has a radius of 100.6 meters, and (while spinning) one rotation takes 314 seconds.
Samuel gets into a carriage at the bottom of the Ferris wheel. At \(t=0\), the wheel starts spinning, and it spins continuously for 736 seconds until an abrupt stop. At that moment, Samuel drops his keys, which fall directly down to the ground.
How far, in meters, are the keys from where Samuel entered the carriage? The tolerance is \(\pm 0.1\) meters.
Solution
Since Samuel got on at the bottom of the wheel, he begins at the midline (horizontally). Thus, we can use a stretched, but unshifted, sine wave to model Samuel’s horizontal position over time.
\[x~=~A\cdot\sin(Bt)\]
The amplitude will match the radius of the wheel.
\[A=100.6\]
To get \(B\), the angular frequency, we divide \(2\pi\) by the period (which was given to us).
\[B~=~\frac{2\pi}{P}\]
\[B~=~\frac{2\pi}{314}\]
\[B~\approx~0.0200101\]
Notice,
\[x~=~100.6\cdot\sin(0.0200101\cdot 736)\]
\[x~=~83.6343837\]
We only care about the absolute value of \(x\).
\[|x| ~=~ 83.6343837\]
Question
A Ferris wheel has a radius of 103.5 meters, and (while spinning) one rotation takes 634 seconds.
Samuel gets into a carriage at the bottom of the Ferris wheel. At \(t=0\), the wheel starts spinning, and it spins continuously for 1500 seconds until an abrupt stop. At that moment, Samuel drops his keys, which fall directly down to the ground.
How far, in meters, are the keys from where Samuel entered the carriage? The tolerance is \(\pm 0.1\) meters.
Solution
Since Samuel got on at the bottom of the wheel, he begins at the midline (horizontally). Thus, we can use a stretched, but unshifted, sine wave to model Samuel’s horizontal position over time.
\[x~=~A\cdot\sin(Bt)\]
The amplitude will match the radius of the wheel.
\[A=103.5\]
To get \(B\), the angular frequency, we divide \(2\pi\) by the period (which was given to us).
\[B~=~\frac{2\pi}{P}\]
\[B~=~\frac{2\pi}{634}\]
\[B~\approx~0.0099104\]
Notice,
\[x~=~103.5\cdot\sin(0.0099104\cdot 1500)\]
\[x~=~77.8472987\]
We only care about the absolute value of \(x\).
\[|x| ~=~ 77.8472987\]
Question
A Ferris wheel has a radius of 67.2 meters, and (while spinning) one rotation takes 456 seconds.
Samuel gets into a carriage at the bottom of the Ferris wheel. At \(t=0\), the wheel starts spinning, and it spins continuously for 939 seconds until an abrupt stop. At that moment, Samuel drops his keys, which fall directly down to the ground.
How far, in meters, are the keys from where Samuel entered the carriage? The tolerance is \(\pm 0.1\) meters.
Solution
Since Samuel got on at the bottom of the wheel, he begins at the midline (horizontally). Thus, we can use a stretched, but unshifted, sine wave to model Samuel’s horizontal position over time.
\[x~=~A\cdot\sin(Bt)\]
The amplitude will match the radius of the wheel.
\[A=67.2\]
To get \(B\), the angular frequency, we divide \(2\pi\) by the period (which was given to us).
\[B~=~\frac{2\pi}{P}\]
\[B~=~\frac{2\pi}{456}\]
\[B~\approx~0.0137789\]
Notice,
\[x~=~67.2\cdot\sin(0.0137789\cdot 939)\]
\[x~=~23.9433894\]
We only care about the absolute value of \(x\).
\[|x| ~=~ 23.9433894\]
Question
A Ferris wheel has a radius of 62.7 meters, and (while spinning) one rotation takes 115 seconds.
Samuel gets into a carriage at the bottom of the Ferris wheel. At \(t=0\), the wheel starts spinning, and it spins continuously for 248 seconds until an abrupt stop. At that moment, Samuel drops his keys, which fall directly down to the ground.
How far, in meters, are the keys from where Samuel entered the carriage? The tolerance is \(\pm 0.1\) meters.
Solution
Since Samuel got on at the bottom of the wheel, he begins at the midline (horizontally). Thus, we can use a stretched, but unshifted, sine wave to model Samuel’s horizontal position over time.
\[x~=~A\cdot\sin(Bt)\]
The amplitude will match the radius of the wheel.
\[A=62.7\]
To get \(B\), the angular frequency, we divide \(2\pi\) by the period (which was given to us).
\[B~=~\frac{2\pi}{P}\]
\[B~=~\frac{2\pi}{115}\]
\[B~\approx~0.0546364\]
Notice,
\[x~=~62.7\cdot\sin(0.0546364\cdot 248)\]
\[x~=~52.1381397\]
We only care about the absolute value of \(x\).
\[|x| ~=~ 52.1381397\]
Question
A Ferris wheel has a radius of 39.4 meters, and (while spinning) one rotation takes 89 seconds.
Samuel gets into a carriage at the bottom of the Ferris wheel. At \(t=0\), the wheel starts spinning, and it spins continuously for 226 seconds until an abrupt stop. At that moment, Samuel drops his keys, which fall directly down to the ground.
How far, in meters, are the keys from where Samuel entered the carriage? The tolerance is \(\pm 0.1\) meters.
Solution
Since Samuel got on at the bottom of the wheel, he begins at the midline (horizontally). Thus, we can use a stretched, but unshifted, sine wave to model Samuel’s horizontal position over time.
\[x~=~A\cdot\sin(Bt)\]
The amplitude will match the radius of the wheel.
\[A=39.4\]
To get \(B\), the angular frequency, we divide \(2\pi\) by the period (which was given to us).
